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Solution - Least common multiple (LCM) by prime factorization

16,335
16,335

Step-by-step explanation

1. Find the prime factors of 11

11 is a prime factor.

2. Find the prime factors of 15

Tree view of the prime factors of 15: 3 and 5

The prime factors of 15 are 3 and 5.

3. Find the prime factors of 27

Tree view of the prime factors of 27: 3, 3 and 3

The prime factors of 27 are 3, 3 and 3.

4. Find the prime factors of 121

Tree view of the prime factors of 121: 11 and 11

The prime factors of 121 are 11 and 11.

5. Find the prime factors of 165

Tree view of the prime factors of 165: 3, 5 and 11

The prime factors of 165 are 3, 5 and 11.

6. Build a prime factors table

Determine the maximum number of times each prime factor (3, 5, 11) occurs in the factorization of the given numbers:

Prime factorNumber11 15 27 121 165 Max. occurrence
3013013
5010011
11100212

The prime factor 5 occurs one time, while 3 and 11 occur more than once.

7. Calculate the LCM

The least common multiple is the product of all factors in the greatest number of their occurrence.

LCM = 33351111

LCM = 335112

LCM = 16,335

The least common multiple of 11, 15, 27, 121 and 165 is 16,335.

Why learn this

The least common multiple (LCM), sometimes called the lowest common multiple or least common divisor, is helpful for understanding the relationships between numbers. For example, if it takes Earth 365 days to orbit the sun and it takes Venus 225 days to orbit the sun and both are in perfect alignment at the time this scenario is given, how long will it take for Earth and Venus to align again? We can use LCM to determine that the answer would be 16,425 days.

LCM is also a very important part of many mathematical concepts that also have real-world applications. For example, we use LCMs when adding and subtracting fractions, which we use quite frequently.